Consider an integral domain $R$ and a short exact sequence of left-modules $$ 0\rightarrow A \rightarrow B \rightarrow C \rightarrow 0. $$
I want to understand the relation between their torsion sub-modules. I want to understand what is the next term of the following sequence $$ 0\rightarrow A_{tor} \rightarrow B_{tor} \rightarrow C_{tor} \rightarrow ?? $$
In the commutative case, let $K$ be the fraction field of $R$. Then you have for any module $A$, an exact sequence $0\to A_{\mathrm{tor}}\to A\to A\otimes_R K\to A_1\to 0$, where $A_1$ is defined by the above sequence. Then, by snake lemma, you get an exact sequence, $0\to A_{\mathrm{tor}}\to B_{\mathrm{tor}}\to C_{\mathrm{tor}}\to A_1\to B_1\to C_1\to 0$.